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This book confronts the issue of how young people can find a way into the world of algebra. It represents multiple perspectives which include an analysis of situations in which algebra is an efficient problem-solving tool, the use of computer-based technologies, and a consideration of the historical evolution of algebra. The book emphasizes the situated nature of algebraic activity as opposed to being concerned with identifying students' conceptions in isolation from problem-solving activity.
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PERSPECTIVES ON SCHOOL ALGEBRA
Mathematics Education Library VOLUME 22 !"#"$%#$ &'%()* A.J. Bishop, !)#"+, -#%./*+%(01 !/23)4*#/1 54+(*"2%"
&'%()*%"2 6)"*' H. Bauersfeld, 6%/2/7/2'1 8/*9"#0 J. Kilpatrick, 5(,/#+1 -:;:5: G. Leder, !/23)4*#/1 54+(*"2%" S . Turnau, / can deduce that x =
the scribe actually assumes a false length, 1 3 = = . He calculates the –
=
=
x0 =
5
(because
. d . In contrast,
which gives a false width false diagonal using
= I ) . The proportional argument under-
lying the procedure leads the scribe to calculate the inverse of arid to multiply this inverse by the given diagonal d=40. This problem clearly shows the functioning of the ancient false position method and will suffice to elaborate our historical reconstruction of the transition from arithmetic to algebra in Mesopotamian land. However, before going on to our next
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L.G. RADFORD
stop in our historical journey, we need to make the following cultural and epistemological remark: the idea of using a 'false quantity' to start the false position method. leans on a deeper and more complex idea: at the beginning of the problem. the 'true quantity' (i.e. the exact solution of the problem) may be legitimately thought of (,*)4$, another quantity. 'False quantities' thus appear as 9/("D,)*+ of 'true quantities'. Furthermore, this is not a phenomenon restricted solely to mathematics: Mesopotamian thinking is full of metaphors. Odes. epic poems, literary and religious texts, for instance, show an intricate system of metaphorical expressions (see e.g. Wilson. 1901). Algebra, we shall suggest. was couched in such a system. ALGEBRAIC THINKING AS A METAPHOR OF THE FALSE POSITION METHOD As we shall see in later sections of this chapter, where we focus on some technical details, the influence of false position methods in the emergence of algebraic ideas can be discerned through some important structural similarities between false position reasoning and early algebraic thinking. One of the studies of the ancient connection between the Babylonian false position method and algebra was made by François Thurcau-Dangin (1938a). Following the trends of the old interpretation of Babylonian mathematics based on the possibility of translating the calculations shown in many of the tablets into modern algebraic symbolism, he noted a strong parallelism between the calculations done in some problems solved by false position methods and those of the modern algebraic methods6. He then claimed that, indeed, some Babylonian procedures were algebraic procedures. Thureau-Dangin’s main idea was supposedly supported by the fact that, in some problems, the scribe takes the number )#/ as the false solution (such an example could be the problem discussed at the end of the previous section) and when, according to Babylonian procedures. we replace the number )#/ by our modern unknown ‘x’, the problemsolving procedures look much like the modern algebraic procedures. He, (as well as others. e.g. Vogel, 1960), claimed that the number one was actually taken as a */D*/+/#("(%)# of the unknown and i f we cannot. straight out, see the unknown, it is simply because the scribes did not have a symbol with which they could represent it. However, the idea that Babylonians dev